Darboux Transformation Examples

Example 1: 7 Soliton Collision

xᵣ = -10=>10
T = 20
seed = "0"
box = Box(xᵣ, T, Nₓ=500, Nₜ = 500)
λ = [-0.45 + 0.775im, -0.35 + 0.8im, -0.25 + 0.825im, 0.85im, 0.25 + 0.875im, 0.35 + 0.9im, 0.45 + 0.925im]
xₛ = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
tₛ = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

calc = Calc(λ, tₛ, xₛ, seed, box)

solve!(calc)
heatmap(calc)

[ Info: Initializing Box with 1 period(s) and dx = 0.0, Nₜ = 500.
[ Info: Longitudinal range is [-10, 10], transverse range is [-10.0, 10.0)
[ Info: Done computing t, x, ω
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (7,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (6,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (5,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (4,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (3,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (2,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,2)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (2,2)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,3)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (3,2)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (2,3)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,4)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (4,2)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (3,3)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (2,4)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,5)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (5,2)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (4,3)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (3,4)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (2,5)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,6)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (6,2)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (5,3)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (4,4)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (3,5)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (2,6)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,7)
/home/runner/.julia/packages/GR/G9I5v/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS

Example 2: Fifth Order Maximal Intensity Breather

xᵣ = -10=>10
λ₁ = 0.98im
λ, T, Ω = params(λ = λ₁)
box = Box(xᵣ, T, Nₓ=500, Nₜ = 500, n_periods = 3)

λ = λ_maximal(λ₁, 5) # array of 5 eigenvalues
xₛ = [0.0, 0.0, 0.0, 0.0, 0.0]
tₛ = [0.0, 0.0, 0.0, 0.0, 0.0]

seed = "exp"
calc = Calc(λ, tₛ, xₛ, seed, box)

solve!(calc)
surface(calc)

[ Info: Passed λ=0.0 + 0.98im, computed T = 15.787097084991364 and Ω = 0.3979949748426484
[ Info: Initializing Box with 3 period(s) and dx = 0.0, Nₜ = 500.
[ Info: Longitudinal range is [-10, 10], transverse range is [-23.680645627487046, 23.680645627487046)
[ Info: Done computing t, x, ω
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (5,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (4,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (3,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (2,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,2)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (2,2)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,3)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (3,2)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (2,3)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,4)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (4,2)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (3,3)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (2,4)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,5)
/home/runner/.julia/packages/GR/G9I5v/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS

Example 3: 3 Soliton Collision on a $cn$ background

xᵣ = -10=>10
T = 20

box = Box(xᵣ, T, Nₓ=500, Nₜ = 500, n_periods = 1)
λ = [-0.3+0.85im, 0.9im, 0.3+0.95im]
xₛ = [0.0, 0.0, 0.0]
tₛ = [0.0, 0.0, 0.0]

seed = "cn"
calc = Calc(λ, tₛ, xₛ, seed, box, m = 0.5)

solve!(calc)
surface(calc)

[ Info: Initializing Box with 1 period(s) and dx = 0.0, Nₜ = 500.
[ Info: Longitudinal range is [-10, 10], transverse range is [-10.0, 10.0)
[ Info: Done computing t, x, ω
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (3,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (2,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,1)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,2)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (2,2)
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,3)
/home/runner/.julia/packages/GR/G9I5v/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS

Example 4: First Order Breather matched to a $dn$ Background

xᵣ = -10=>10
m = 2/5
λ = λ_given_m(m, q=4)
λ, T, Ω = params(λ = λ, m=m)
box = Box(xᵣ, T, Nₓ=500, Nₜ = 500, n_periods = 3)

λ = λ_maximal(λ, 1, m=m)
xₛ = [0.0]
tₛ = [0.0]

seed = "dn"
calc = Calc(λ, tₛ, xₛ, seed, box, m=m)

solve!(calc)
heatmap(calc)

[ Info: Passed λ=0.0 + 0.8588568879615993im, computed T = 14.220154971930045 and Ω = 0.4418506914715286
[ Info: Initializing Box with 3 period(s) and dx = 0.0, Nₜ = 500.
[ Info: Longitudinal range is [-10, 10], transverse range is [-21.330232457895068, 21.330232457895068)
[ Info: Done computing t, x, ω
[ Info: Calculating Lax pair generating functions rₙₚ(x,t) and sₙₚ(x,t) for (n,p) = (1,1)
/home/runner/.julia/packages/GR/G9I5v/src/../deps/gr/bin/gksqt: error while loading shared libraries: libQt5Widgets.so.5: cannot open shared object file: No such file or directory
connect: Connection refused
GKS: can't connect to GKS socket application

GKS: Open failed in routine OPEN_WS
GKS: GKS not in proper state. GKS must be either in the state WSOP or WSAC in routine ACTIVATE_WS